Why can I write $\mathbb E[\mathbb E[X|A]1_{A}|B]=\mathbb E[X|A]P(A|B)$

Question

I know that $\mathbb E[XY]= \mathbb E[X]\mathbb E[Y]$ if $X$ and $Y$ are independent.

Question: Let $A,B$ be two given events, why can I write $\mathbb E[\mathbb E[X|A]1_{A}|B]=\mathbb E[X|A]P(A|B)$

surely in order to "separate" $E[X|A]$ and $1_{A}$ under expectation, we need to know that the random variables are independent but both are by definition dependent on $A$, so how can I be sure that $E[X|A]$ and $1_{A}$ are independent?

Further, if we can separate them, then why is $\mathbb E[ \mathbb E[X|A] |B]=\mathbb E[X|A]$

Is it because $E[X|A]$ is a constant?

Answer 1

Note that $E[X|A]$ is a constant and hence $$ E[E[X|A]1_{A}|B]=E[X|A]\cdot E[1_{A}|B]=E[X|A]P(A|B), $$ where the first equality follows from the linearity of conditional expectation with respect to an event. You do not need to use independence here.